A model-free design of reduced-order controllers and application to a DC servomotor

Automatica 50 (2014) 2142–2149

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Brief paper

A model-free design of reduced-order controllers and application to a DC servomotor✩ Sofiane Khadraoui a , Hazem Nounou a,1 , Mohamed Nounou b , Aniruddha Datta c , Shankar P. Bhattacharyya c a

Department of Electrical and Computer Engineering, Texas A&M University at Qatar, Doha, Qatar

b

Department of Chemical Engineering, Texas A&M University at Qatar, Doha, Qatar

c

Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA

article

info

Article history: Received 6 January 2013 Received in revised form 27 February 2014 Accepted 22 April 2014 Available online 18 June 2014 Keywords: Model-free control Unknown control architecture Reference model design Low-order control design Controller tuning Frequency response Performance achievement Control of DC servomotor

abstract This paper presents a new model-free technique to design fixed-structure controllers for linear unknown systems. In the current control design approaches, measured data are used to first identify a model of the plant, then a controller is designed based on the identified model. Due to errors associated with the identification process, degradation in the controller performance is expected. Hence, we use the measured data to directly design the controller without the need for model identification. Our objective here is to design measurement-based controllers for stable and unstable systems, even when the closed-loop architecture is unknown. This proposed method can be very useful for many industrial applications. The proposed control methodology is a reference model design approach which aims at finding suitable parameter values of a fixed-order controller so that the closed-loop frequency response matches a desired frequency response. This reference model design problem is formulated as a nonlinear programming problem using the concept of bounded error, which can then be solved to find suitable values of the controller parameters. In addition to the well-known advantages of data-based control methods, the main features of our proposed approach are: (1) the error is guaranteed to be bounded, (2) it enables us to avoid issues related to the use of minimization methods, (3) it can be applied to stable and unstable plants and does not require any knowledge about the closed-loop architecture, and (4) the controller structure can be selected a priori, which means that low-order controllers can be designed. The proposed technique is experimentally validated through a real position control problem of a DC servomotor, where the results demonstrate the efficacy of the proposed method. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Traditional approaches dealing with the modeling, simulation and control of physical systems require accurate description of their dynamic characteristics. Mathematical models of such systems can be obtained using basic physical laws. However, in many practical applications, it is often difficult to obtain an

✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Jun-ichi Imura under the direction of Editor Toshiharu Sugie. E-mail addresses: [email protected] (S. Khadraoui), [email protected] (H. Nounou), [email protected] (M. Nounou), [email protected] (A. Datta), [email protected] (S.P. Bhattacharyya). 1 Tel.: +974 4423 026; fax: +974 4423 0064.

http://dx.doi.org/10.1016/j.automatica.2014.06.001 0005-1098/© 2014 Elsevier Ltd. All rights reserved.

accurate model from basic principles because of the complexity and lack of complete knowledge about the system dynamics. To overcome such a difficulty, mathematical models are built on the basis of experimental data obtained from experiments carried out on the plant. The use of such empirical models is very attractive for control design due to the simplicity of their derivation using system identification. However, the identification of physical systems usually assumes prior knowledge of the model structure or order, which is often unavailable or subject to uncertainties. Such uncertainties and unavailable information increase the difficulty of obtaining simple and reliable models for control design purposes. Hence, errors associated with such models result in degradation of the performance of the designed controller (Athans, Rohrs, Valavani, & Stein, 1982; Ioannou & Kokotovic, 1984). Due to the fact that in model-based methods, the controller performance is dependent on the accuracy of mathematical

S. Khadraoui et al. / Automatica 50 (2014) 2142–2149

models, data-based control design approaches are considered as an attractive alternative. It has been shown in many practical control applications that controllers can directly be designed using only a set of measurements given either in the timedomain or in the frequency-domain. A recent survey on databased control methods can be found in Hou and Wang (2013), where the authors discussed in detail the relationship and differences between model-based and data-based methods. In time-domain approaches, data-based control design methods for reference and tracking problems have been presented in Bazanella, Campestrini, and Eckhard (2012), Campi, Lecchini, and Savaresi (2002), Hjalmarsson, Gevers, Gunnarsson, and Lequin (1998), Karimi, Miskovic, and Bonvin (2004) and Spall and Cristion (1998). There are also other model-free control methods in time-domain, such as lazy learning methods (Bertolissi, Birattari, Bontempi, Duchteau, & Bersini, 2002; Kobayashi, Konishi, & Ishigaki, 2007; Takao, Yamamoto, & Hinamoto, 2004) and data-based adaptive control methods (Bertolissi et al., 2002; Hou & Jin, 2011; Zhu & Sun, 2012). Most time-domain data-based approaches outlined above are relatively simple and easy. However, the main difficulty with such techniques is that insights on closed-loop stability are not transparent. Additionally, most of them are iterative and usually do not guarantee optimal global solutions. Unlike the time domain, frequency domain is generally viewed as a favorable domain for interpreting the plant dynamics and formulating controller design objectives. This fact is due to the useful information that frequency-domain techniques can provide such as phase and gain stability margins, frequency bandwidth, and resonant frequency. Another feature of measurement-based techniques developed in the frequency domain is that they allow us to measure the robustness of the designed controllers. These facts enable data-based control design process in the frequencydomain to be efficient, more transparent, and very successful. Frequency-domain data-based control techniques are often based on the frequency response of the plant which is assumed to be available or can be measured experimentally. Such an assumption is often valid in many applications. Based on these frequency response data, it is possible to design controllers without using any mathematical model of the plant. In Keel and Bhattacharyya (2008), it has been shown that the set of all stabilizing PID controllers, achieving a desired gain and phase margin or infinitynorm constraints, can be obtained using only the frequencydomain data. The quantitative feedback theory (QFT) approach which is based on loop-shaping in the Nichols chart has been presented in Horowitz (1993). Based on plant frequency response data, the authors in den Hamer (2010) and Karimi and Galdos (2010) proposed a method to design fixed-structure controllers such that the infinity-norm of some weighted transfer functions in the closed-loop system is minimized. The main limitation in the approach presented in den Hamer (2010) and Karimi and Galdos (2010) is the selection of the weighting functions to establish the required performance. The choice of such weighting functions is generally cumbersome, and often based on trial-anderror methods. Moreover, such a method assumes that controllers to be designed are linearly parameterized (the controller transfer function is linear with respect to the parameters) which is not the case in many practical applications. The authors in Garcia, Karimi, and Longchamp (2006) used frequency-domain data to tune PID controllers that achieve performance specifications in terms of either gain and phase margins or infinity-norm of the sensitivity function and the complementary sensitivity function. In this latter, the design problem has been formulated as a nonlinear nonconvex optimization problem, in which a local optimization method has been proposed for its solution. However, the use of local optimization methods typically leads to the convergence to a

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local minimum of the objective function. Another limitation of the technique discussed in Garcia et al. (2006) is that its applicability requires that the Hessian matrix of the objective function, which depends on the number of the controller parameters, must be nonsingular (positive definite matrix). Moreover, when singularities occur in the objective function, its gradient does not exist, and hence the optimum solution may not be attainable. The authors in Datta et al. (2013) recently proposed a measurement-based modeling technique for unknown systems, which has also been used to design controllers. In many practical control applications, the closed-loop system consists of a set of linear subsystems connected together in a certain complex configuration. In practical engineering applications, such subsystems generally include electrical, mechanical, thermal, hydraulic and pneumatic elements such as motors, pumps, hydraulic cylinder, valves, reduction gears, amplifiers, cables, etc. The complexity of interconnection between subsystems generally increases the difficulty of identifying the overall closed-loop architecture. In such a situation, model-based methods may not be suitable, and hence data-based control techniques can be used. Nevertheless, data-based control design approaches available in the literature assume generally a simple and known closed-loop scheme such as unity and non-unity feedback control schemes and feedforward/feedback control scheme. In such simple schemes, the closed-loop relationship, which depends on the controller and elements constituting the closed-loop system, can easily be derived. However, when the closed-loop architecture (interconnection between subsystems) is either unknown or difficult to identify (available but complicated), it is relatively difficult to find the closed-loop relationship in closed-form that defines the closedloop system dependence of the controller and the other elements (subsystems) of the plant. Therefore, the application of the existing data-based control design methods becomes challenging. In such a situation, where the closed-loop relationship cannot explicitly be constructed, a possible way to be able to apply the available databased control design techniques can be achieved by embedding first any stabilizing controller in the complicated closed-loop configuration. The resulting stable closed-loop system is considered as the open-loop system (inner loop) to be used for collecting data. Such data are then utilized to design a second feedback controller introduced in an outer loop. Unlike such a solution, we aim in this paper to address the above problem by proposing an approach to design one-degree-of-freedom controller for linear time-invariant SISO systems using measured data collected from an unknown or complicated closed-loop configuration. Such an approach is useful in practical applications, where the control objective is often to update an actual controller with a new controller that achieves better or different performance measures. The proposed method relies on a general relationship relating the closed-loop frequency response to that of the controller, which can be derived by either using a Linear Fractional Transformation or a Multilinear Form Lemma. The derivation of such a closed-loop relationship is based on performing three sets of experiments. Our proposed control method is based on the concept of reference model design, in which we formulate the control problem as follows: for a selected controller structure, the set of admissible values of the controller parameters is computed such that the error modulus, between the closedloop frequency response and the desired frequency response at each fixed frequency (from the frequency range of operation) is bounded by a small quantity. The main advantage of such a proposed method with respect to other methods is its ability to design one-degree-of-freedom controllers for stable and unstable plants, as well as its applicability even when the closed-loop control architecture is complicated or unknown. Another feature of our proposed method is that it does not need the use of any minimization methods. Moreover, no knowledge of the model order or structure

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the closed-loop frequency response, H (jω), to that of the controller, C (jω), at each frequency ω can be derived from (2) as follows: H (jω) =

Fig. 1. Interconnected unknown linear system with the controller.

is required in the design process. The idea is based on the fact that all relevant information about the system to be controlled is included in the set of measurements (measured frequency response). The paper is organized as follows. In Section 2, the measurement-based modeling approach corresponding to the computation of the closed-loop system frequency response is presented. Section 3 is dedicated to the design of fixed structure/order controllers based on measurements at a finite set of frequencies. In Section 4, we demonstrate the practical applicability of the proposed method to control the angular position of a DC servomotor. Finally, some concluding remarks and future research directions are outlined in Section 5. 2. Controller dependence on the closed-loop frequency response In this section, we aim to derive the controller dependence on the closed-loop frequency response, when the closed-loop system architecture is unknown or difficult to identify. As outlined above in the introduction section, the closed-loop system scheme in practical applications is made up of several linear subsystems interconnected in a complicated configuration. In such a complex (or unknown) closed-loop system configuration, the controller is placed at a particular fixed location, but its connection to the plant subsystems may be unknown. It is well known that any simple or complicated closed-loop diagram can be represented, using Linear Fractional Transformation (LFT), by a system, P, interconnected with a controller, C (s), as shown in Fig. 1. In this contribution, the plant P, which contains all subsystems of the closed-loop system, is considered to be an unknown linear SISO system. Such a plant, P, is assumed to be unknown because we consider here the case of complicated (or unknown) subsystem interconnection (closedloop architecture). The design problem considered here can be viewed as finding a functional dependence of the controller, C (s), on the frequency response of the closed-loop system linking the input yr (the desired output trajectory) to the output y. This closed-loop relationship can be obtained using Linear Fractional Transformation. From Fig. 1, we can write: y P (s) = 11 z P21 (s)

 



P12 (s) P22 (s)

  yr x

(1)

x = C (s)z , where P11 (s), P12 (s), P21 (s), and P22 (s) are fixed transfer functions. In the existing control design approaches (data-based or modelbased methods), where the closed-loop architecture is known, P11 (s), P12 (s), P21 (s), and P22 (s) are known fixed transfer functions. However, in our case where the closed-loop architecture is assumed to be unknown (or difficult to identify), such transfer functions are fixed but unknown. Based on Eq. (1), the closedloop transfer function from the input, yr , to the output, y, can be expressed as follows: −1 −1 P11 P22 + P12 P21 P22 − P11 C



H (s) =

−1 P22 −C



.

(2)

−1 Let us define A(s) = −P11 (s)P22 (s), B(s) = P11 (s) − P12 (s)P21 (s) −1 −1 P22 (s) and D(s) = −P22 (s). Hence, the relationship that relates

A(jω) + B(jω)C (jω) D(jω) + C (jω)

,

(3)

where the three unknown terms A(jω), B(jω) and D(jω) are fixed for any given frequency ω. Performing three experiments is enough to solve for the parameters A(jω), B(jω) and D(jω) by embedding any three stabilizing controllers C1 (s), C2 (s) and C3 (s), in the control system scheme (see Fig. 1) and then measuring the respective closed-loop frequency responses, H1 (jω), H2 (jω) and H3 (jω), over a finite set of frequencies ω ∈ {ω1 , ω2 , . . . , ωN }, where N is an integer to be chosen by the used. In each experiment i (i = 1, 2, 3), the stabilizing controller, Ci (s), i = 1, 2, 3, is first implemented, then a harmonic analysis is carried out on the closed-loop system by applying a sinusoidal reference signal yr of a given amplitude and different frequencies ω ∈ {ω1 , ω2 , . . . , ωN } and recording the resulting output signal y. The frequency response Hi (jω) (for ω ∈ {ω1 , ω2 , . . . , ωN }) can then be obtained from these input/output measurements using Fourier analysis. The set of frequencies ω ∈ {ω1 , ω2 , . . . , ωN } is similar for the three experiments and should be chosen carefully depending on the frequency range of operation of the controlled system. Indeed, ω1 is a very low frequency for which the steady-state response can be obtained, while ωN is a high frequency for which the output signal magnitude tends to zero. Splitting the frequency range of interest, [ω1 , ωN ], into a finite set of equally spaced frequencies can be viewed as a natural and simple way to select intermediate frequencies, ω2 , . . . , ωN −1 . The selection of too narrow frequency spaces, which results in a large number of frequencies, N, allows us to extract a good information about the closed-loop frequency responses Hi (jω), i = 1, 2, 3. However, the use of a high number of frequencies is generally time consuming (a large computational burden) and undesirable from a practical point of view. On the other hand, the selection of too large frequency spaces leads to a small set of frequencies, which therefore may have a significant impact on the quality of the derived closed-loop frequency response. The use of inaccurate frequency responses results in degradation of the controller performance, or even in loss of closed-loop stability. The set of frequencies should be limited, but sufficient in order to obtain an appropriate and meaningful frequency response. A systematic way for selecting such a limited number, N, and therefore reducing the computational burden, can be achieved using unequally spaced frequencies (different frequency increments). Hence, fine frequency increments are usually needed near the ranges of frequencies that exhibit rapid changes. The use of such fine frequency increments around resonance and antiresonance phenomena allows us to derive a frequency response with high resolution. Over the ranges of frequencies where the variations of the frequency response is very slow, larger frequency increments can be selected. Remark 1. When the objective is to design a two or more degreeof-freedom controller, it is often difficult to find the closed-loop relationship-dependent controller using Linear Fractional Transformation due to the complexity. In such a case, the Multilinear Form Lemma (Bode, 1945), which can be viewed as a generalization of the Linear Fractional Transformation result, can be used. Lemma 1 (Multilinear Form Lemma). Let the matrices A, K ∈ Rn×n and the vectors b, f ∈ Rn be given. Let p = (p1 , p2 , . . . , pn ) ∈ Rn be a parameter vector. Denoting the ith row of a matrix M as (M )i , suppose that the matrix A¯ (p) and the vector b¯ (p) are defined as

(A¯ (p))i = [ai1 , ai2 , . . . , ain ] + pi [ki1 , ki2 , . . . , kin ], (b¯ (p))i = bi + pi fi .

(4)

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Then, for the linear system A¯ (p)x(p) = b¯ (p), each component solution x(p) (for m = 1, 2, . . . , n) can be expressed by a multilinear rational function of the parameters pi as follows:

α0m + xm (p) =



αim pi +



βim pi +



i

β0m +

 i

i,j i,j

αijm pi pj + · · · βijm pi pj + · · ·

.

(5)

where the real, Re(ωk ), and imaginary parts, Im(ωk ), are functions of A(jωk ), B(jωk ), D(jωk ) and C (jωk ). Assume that the frequency response corresponding to the desired closed-loop system (the so called reference model), denoted by H ∗ (s), can be written as follows: H ∗ (jωk ) = Re∗ (ωk ) + jIm∗ (ωk ).

Proof. The proof is based on Cramer’s rule applied to a linear system of equations and the properties of determinants. In the case of designing one degree-of-freedom controller (i = 1), it is clear that the derivation of Eq. (3) using Eq. (5) is straightforward by considering the variable xm as the closed-loop frequency response and the parameter p as the controller, and the terms A, B and D depend on the coefficients α1m , α0m , β0m and β1m .

(8)

where Re (ωk ) and Im (ωk ) are respectively the real and imaginary parts of the desired frequency response that describes the desired closed-loop performance. Hence, our model reference design problem consists of finding the controller parameter, θ , so that the closed-loop frequency response H (jωk ), expressed by (7), best fits the desired frequency response H ∗ (jωk ) defined by (8) over all frequencies ω1 , ω2 , . . . , ωN . Let e(ωk ) ∈ C be the error between the computed frequency response H (jωk ) and the desired frequency response H ∗ (jωk ), at the frequency ωk , k = 1, 2, . . . , N. It is defined as follows: ∗

where, for l = 0, 1, . . . , the coefficients αlm depend on the elements of A, K , b, f , whereas the coefficients βlm depend only on A, K .

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∗

3. Controller design

e(ωk ) = H ∗ (jωk ) − H (jωk ).

The main objective of this section is to design a measurementbased fixed-structure controller which guarantees some performance measures for the closed-loop system given in Fig. 1. In addition to the desired closed-loop performance measures, it is desired here to design low-order controllers for simplicity of implementation, especially for embedded systems. The complexity of the method proposed here with respect to the existing databased control design approaches is related to the workload needed to conduct three experiments required for the design of the controller. Indeed, the time spent on collecting measured data in our proposed method is typically three times the time needed to collect measurements in the existing frequency-domain data-based control methods. However, its main advantage is that no knowledge about the closed-loop architecture is required. Another feature is that it can be applied to stable and unstable systems. Such features of the proposed technique can be useful in the control of industrial applications. To design the measurement-based controller, let us consider a fixed-structure controller C (s) defined as follows:

Our design problem can be reformulated as follows: find suitable values of θ so that for any fixed frequency, ωk , k = 1, 2, . . . , N, the error modulus remains bounded by a quantity e, as follows:

p 

C (s) =

l =0 m



βl sl .

(6)

αi si

i =0

where αi and βl (for i = 0, . . . , m and l = 0, . . . , p) are the controller parameters, with m ≥ p in order to ensure the causality of the controller. Indeed, the selection of large values for m and p (which means that a high-order controller is designed) can provide improved compensation capabilities. However, the fact that such high-order controllers are time consuming limits their utilization in embedded microcontrollers. Hence, in many practical applications, low-order controllers (such as PI, PID controllers, etc.) are generally selected to simplify the implementation step. Let θ = [α0 , . . . , αm , β0 , . . . , βp ] be a vector containing the controller parameters. Using the conceptual framework of reference model, our control design problem can be stated as follows: find the value of θ for which the closed-loop frequency response is close enough to a desired closed-loop frequency response denoted by H ∗ (jωk ), which meets some desired closed-loop performance measures, over the selected range of frequencies, ωk , k = 1, . . . , N. Using the frequency response of the controller given in (6), the closed-loop frequency response H (jωk ) expressed in (3) can be rewritten for all frequencies considered ωk ∈ {ω1 , ω2 , . . . , ωN } as follows: H (jωk ) = Re(ωk ) + jIm(ωk ).

(7)

|(e(ωk ))| ≤ e,

∀k = 1, 2, . . . , N ,

(9)

(10)

where the bound e > 0 is a positive real number to be chosen small enough. Such a bound e¯ can be viewed as a tuning parameter that allows us to improve the quality of fit between the closedloop and desired frequency responses. Indeed, the selection of a bound e¯ very close to zero allows us to ensure that the frequency responses H (jωk ) and H ∗ (jωk ) are quite equal at each frequency ωk , k = 1, 2, . . . , N. However, finding a solution for the above problem with such a choice might not be feasible. On the other hand, relaxing the control design problem by using a large value for the upper bound e¯ leads to some controller parameter values for which the desired performance may not be achieved. Hence, the choice of such a value e¯ is a compromise between the quality of fit between H (jωk ) and H ∗ (jωk ) and the possibility to find a suitable controller. The control problem (10) is a nonlinear programming problem with N inequality constraints. Solving such a system of nonlinear inequality constraints (10) yields a set of controllers. Let Θ be the solution set of suitable values of the controller parameter θ for which the conditions (10) are fulfilled. Hence, our control design problem is to find Θ so that

    |(e(ω1 ))| ≤ e     |(e(ω ))| ≤ e     2 Θ = θ ∈ D  ..   .     |(e(ω ))| ≤ e  N

(11)

where D is the domain of θ . Remark 2. Without any loss of generality, instead of having one fixed value of the bound e as presented above, it is possible to choose a different value of the bound, ek , for each frequency ωk . The problem (11) is a nonlinear programming problem, where its solution provides a set of controllers that achieve the desired performance. Here, to efficiently solve the above design problem (11), we propose the algorithm summarized in Table 1. The algorithm requires an initial search space D (possibly large) for controller parameters (6). The union of all p points used for the sampling step (step 2 in Table 1) constitutes approximately (depending on the chosen value ϵ ) the entire search space D . In step 3.b, the subset X ⊆ D is the union of all points in the initial search space D for which |e(ωk )| ≤ e¯ is satisfied at one frequency

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Table 1 Algorithm used to solve the model reference control design problem (11). In: frequency set ωk , parameters search space D , error e(ωk ), error bound e¯ , max(¯e), ϵ . Out: solution set of the controller parameters Θ . Begin 1. Initialize Θ = D , 2. Sample the entire space of parameters D at p distinct points using uniformly spaced value ϵ , 3. For each frequency ωk (k = 1, 2, . . . , N) do a. evaluate |e(ωk )| at the p points created in the search space D , b. extract from the search space, the subset X ⊆ D of all points in D having coordinates where |e(ωk )| ≤ e is satisfied, c. Θ = Θ ∩ X , d. If Θ = ∅ then increase the accuracy by reducing ϵ value or increase slightly the bound e ≤ max(e), and go to 1, Else Θ = Θ, Endif Endfor End

ωk fixed by the loop ‘‘for’’. In the case where constraints of the above control design problem are not fulfilled for a given value e¯ , the algorithm allows us to slightly increase the value of such a bound in order to relatively relax such constraints. However, it is worth noting that e¯ should not exceed a maximum value max(¯e) specified by the user, to avoid mismatches between the closed-loop and desired frequency responses. The maximum value max(¯e) is selected such that a satisfactory fit between closed-loop system and reference model can be achieved. The main feature of this algorithm is that it enables us to provide the solution set Θ within which multiple controller parameter values can be selected. Moreover, it allows us to avoid issues related to the minimization methods which may eventually lead to a local minimum. Also, with such an algorithm, it is guaranteed that the error is bounded. However, its main limitation is the computational complexity which increases with the number of the controller parameters. This limitation occurs also when methods based on the minimization of error functions are used. This problem can be avoided when loworder controllers are designed. Remark 3. The application of the algorithm presented in Table 1 provides the solution set Θ corresponding to the controller parameters defined in (6). Hence, any choice of the controller parameters within the resulting solution set Θ will satisfy the above inequalities (11) at each frequency ωk , as well as the desired closed-loop performance. Nevertheless, it is often desirable to select the controller parameters to be away from the boundary of the solution set Θ to avoid any unexpected experimental performance degradation. Remark 4. In the case where the above problem (11) is not feasible, which means that the solution set is empty, i.e. Θ = ∅, the assumed controller structure given in Eq. (6) and/or the parameter search space D must be changed, and the approach can be reapplied using a new (possibly higher order) controller structure and/or different search space D . One important issue related to measurement-based control design is closed-loop system stability. It is worth noting here that bounded error in the frequency response does not necessarily guarantee closed-loop stability. Hence, a posteriori check of closed-loop stability is of a great importance. Closed-loop stability can be analyzed using frequency response data of stable open-loop system. According to results developed in Keel and Bhattacharyya (2010) and Mohsenizadeh, Darbha, Keel, and Bhattacharyya (2012),

Fig. 2. Photograph of a DC servomotor.

a necessary and sufficient condition for closed-loop stability is that the Nyquist contour of the stable open-loop system, traversed in the sense of growing frequencies (ω1 < ω2 < · · · < ωN ), does not encircle the critical point (−1, 0j). When the open-loop system is unstable, stability check can be performed based either on the identification of the closed-loop system model (Sala & Esparza, 2005), or on the estimation of the phase of the closed-loop transfer function (Lanzon, Lecchini, Dehghani, & Anderson, 2006). Unlike measurement-based control design methods that require an a posteriori check of closed-loop stability, including the method proposed here, there are some data-based control approaches that take into account closed-loop stability issues (van Heusden, Karimi, & Bonvin, 2008, 2011). Nevertheless, in such approaches, only a sufficient condition for closed-loop stability is considered, and the designed stabilizing controllers are assumed to be linearly parameterized. In this paper, a posteriori stability check can be viewed as an additional step for the controller design. However, such an additional step is inexpensive, fast and easy. Moreover, for safety reasons, it is sometimes preferable to double check closed-loop stability a posteriori even if the controller is designed a priori to achieve this latter. 4. Application to the position control of DC servomotors This section is devoted to an experimental application of the proposed control design approach. The objective of this application is to design a reduced-order measurement-based controller to control the position of a Quanser DC (Direct Current) servomotor (see Fig. 2). Although, DC servomotors are very simple devices that do not require complexity for their controller design or implementation phases, they are used here only to validate our proposed method. In the literature, depending on the application, it is of interest to control the following output variables: the angular speed ωm or angular position θm . In this application,  we focus on the control problem of the angular position θm = ωm dt. It is worth noting that the studied system (open-loop position system) is unstable since it involves an integrator term. Hence, let us consider the DC servomotor system placed in a unity feedback configuration with a controller as given in Fig. 3. Here, it is clear that the closed-loop system architecture is known. However, the controller design process used here does not require the knowledge of such information. In servomotor applications, it is often expected that the closedloop system meets some design specifications, such as small overshoot and settling time. Hence, we focus here on designing a measurement-based reduced order controller that guarantees

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Fig. 3. Closed-loop control system.

stability and meets the following closed-loop design specifications for a unit step input: closed-loop behavior with small overshoot, settling time ts5% of about 0.1 s, and small steady-state error. Fig. 4. Solution set of suitable controller parameter values.

4.1. Controller design Here, the measurement-based control approach presented in the previous section is utilized to design a reduced-order controller that ensures closed-loop performance requirements outlined above. The frequency of interest considered in this application ranges from 0 to 100 Hz, in which a set of 250 distinct frequencies has been selected. Three constant-gain controllers C1 (s) = 7, C2 (s) = 10 and C3 (s) = 15 stabilizing the closedloop system have been used to perform a harmonic analysis on the three closed-loop systems in order to derive the relationship linking the closed-loop frequency response and that of any used controller. Once the three sets of closed-loop frequency response data are available over the range of frequencies {ω1 , ω2 , . . . , ω250 }, the terms A(jωk ), B(jωk ) and D(jωk ) at each frequency, ωk , k = 1, 2, . . . , 250, can easily be derived. In this practical example, we are interested in designing PD controllers due to its simplicity (two parameters). It is worth noting here that the integral action is not considered in the controller to be designed due to the fact that the plant itself involves an integrator. Let a PD-structured controller C (s) be defined as follows: C (s, θ ) = Kp +

Kd s Kd s 100

+1

.

(12)

where θ = [Kp , Kd ] is a vector containing the proportional and derivative gains of the PD controller. From the desired closed-loop performance specifications, a reference model has been selected to be H ∗ (s) = K /(1 +τ s), where K is chosen to be one in order to ensure zero steady-state error and the time constant which depends on the desired settling time is selected to be τ = ts5% /3 = 0.033 s. Its frequency response can be K −j K τ ω2 k 2 . According to our method expressed as H ∗ (jωk ) = 2 2 1+τ ωk

1+τ ωk

presented in the previous section, the control design problem here is to find the set Θ of suitable values of the controller parameter θ = [Kp , Kd ], so that the following inequality constraints are satisfied for any fixed frequency ωk , k = 1, 2, . . . , 250:



Re(ωk ) −

K 1 + τ 2 ωk2

2



+ Im(ωk ) +

K τ ωk 1 + τ 2 ωk2

2

≤ e,

(13)

where Re(ωk ) and Im(ωk ) are respectively the real and imaginary parts of the closed-loop frequency response which depend on the sets A(jωk ), B(jωk ), D(jωk ) and the selected PD controller parameters. We solve the above nonlinear programming problem to find the solution set Θ of the controller parameter θ = [Kp , Kd ] for which the error modulus (13) is less than or equal to e = 0.05 over the set of selected frequencies. Applying the algorithm presented

in Table 1 to our design problem (13), the solution set shown in Fig. 4 is obtained. The dark colored area Θ corresponds to the set of admissible values Kp × Kd of the PD controller parameters that ensure bounded error as defined in (13) at each frequency ωk , k = 1, 2, . . . , 250. 4.2. Implementation and experimental results analysis The controller C (s) to be implemented is chosen arbitrarily by taking any point (Kp , Kd ) within the solution set Θ . In this application, we will test the following PD controllers: C1 (s) = .525s , and C3 (s) = 19.4 + 0.05.5s . 18 + 0.055.55s , C2 (s) = 18.6 + 0.0525 100

s +1

100

s+1

100

s+1

4.2.1. Closed-loop stability analysis Before experimentally validating the closed-loop response when the controller is implemented (see Fig. 3), it is important to check the stability of the resulting closed-loop system. Since the system studied here is unstable, the frequency response data of the open-loop system is not enough, using the Nyquist criterion, to decide whether the closed-loop system is stable or not. Hence, closed-loop stability test can be performed here using the technique presented in van Heusden et al. (2008, 2011). According to van Heusden et al. (2008, 2011), a sufficient condition for closedloop stability is that δ = supωk |Ms (jωk ) − L(jωk )(1 − Ms (jωk ))| ≤ 1, where L(jωk ) and Ms (jωk ) are respectively the open-loop frequency response obtained for C1 (s), C2 (s), or C3 (s), and the closed-loop frequency response derived using any stabilizing controller (for instance one of the three constant-gain stabilizing controllers). After computation, we obtain δ1 = 0.542 < 1, δ2 = 0.536 < 1, and δ3 = 0.53 < 1, respectively for the designed PD controllers, C1 (s), C2 (s), and C3 (s), which confirm closed-loop stability. From closed-loop models identified for C1 (s), C2 (s), and C3 (s), it has also been verified that such selected PD controllers stabilize the plant (Sala & Esparza, 2005). 4.2.2. Implementation of the controller Each of the above selected PD controllers was placed in the unity feedback configuration as in Fig. 3. Then, a harmonic analysis is performed on each closed-loop system by applying a sinusoidal reference input of amplitude 50° and using the same set of frequencies ωk , k = 1, 2, . . . , 250. The experimental results obtained for controllers C1 (s), C2 (s) and C3 (s) are shown in Fig. 5 with solid red line, dashed blue line and solid black line, respectively. The desired frequency response, H ∗ (jω), is plotted in the same figure with solid blue line. As shown in Fig. 5,

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Fig. 5. Experimental and simulation Nyquist plots of the closed-loop system (H (jω) and H ∗ (jω)).

equal to a small quantity. The main feature of the proposed data-based control design method is that it can be applied to stable and unstable systems as well as to situations when the closed-loop architecture is complicated or unknown. Moreover, the design problem does not require the use of minimization methods. Also, it enables us to design low-order controllers. However, the main limitation of the proposed technique is that three experiments are required for the controller design process. Another limitation is related to the computational complexity that can increase with the number of controller parameters. This latter limitation is also applicable to existing data-based control methods. Experimental results obtained from an application to a DC servomotor have demonstrated the efficacy of the proposed approach. The performance of the proposed approach, however, can be affected by various types of uncertainties, such as variation of the physical parameters, external disturbances and limitations of the used sensors to provide accurate measurements. The acquired measurements used for the design of the controller are often affected by measurement noise. Hence, one of our future research directions will particularly address the issue of databased control design in presence of measurement noise which often contributes to the degradation of the desired performance or even to the instability of the closed-loop system. This problem is a critical issue in many practical applications. Acknowledgments This work was made possible by NPRP grant NPRP09-11532-450 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. References

Fig. 6. Comparison between the experimental and desired closed-loop time responses.

the experimental closed-loop frequency responses and desired frequency response are very close to each other. Fig. 6 shows experimental closed-loop time responses obtained for controllers, C1 (s) (solid red line), C2 (s) (dashed blue line), and C3 (s) (solid black line), when a reference input signal of amplitude, θmr = 50° , is applied. To show a comparison between the experimental and the desired closed-loop responses, we plot in the same figure, the simulated time response of the reference model. It can be seen from such a figure that the applied PD controllers ensure the required design specifications. In fact, the resulting closed-loop responses show zero overshoots, neglected steady state errors, and settling times ts15% ≈ 0.11 s, ts25% ≈ 0.104 s and ts35% ≈ 0.09 s for the controllers C1 (s), C2 (s) and C3 (s), respectively. 5. Conclusions and future work In this paper, a measurement-based approach for tuning reduced-order controllers for unknown systems is presented. The approach is based on frequency-domain data obtained from the closed-loop system. It has been shown that fixed-order controllers can be designed, so that the error modulus, between the closedloop frequency response and a desired frequency response that meets some desired performance specifications, is less than or

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Sofiane Khadraoui received the Engineering and Magister degrees in Automatic Control from Abou Bekr Belkaïd University, Tlemcen, Algeria, in 2004 and 2007, respectively. In 2012, he has been awarded Ph.D. degree in Automatic Control by the University of Franche-Comté, Besançon, France. He is currently a Postdoctoral Research Associate at Texas A&M University at Qatar. His main research interests include robust control design for uncertain systems, measurement-based control design for unknown systems, design of structured controllers, robust stability analysis, compensation of nonlinear phenomena (hysteresis, creep, dead-zone, etc), modeling of structured and unstructured uncertainties, optimization and interval computation.

Hazem Nounou (SM’08) received the B.S. degree (magna cum laude) from Texas A&M University, College Station, in 1995, and the M.S. and Ph.D. degrees from Ohio State University, Columbus, in 1997 and 2000, respectively, all in Electrical Engineering. In 2001, he was a Development Engineer for PDF Solutions, a consulting firm for the semiconductor industry, in San Jose, CA. Then, in 2001, he joined the Department of Electrical Engineering at King Fahd University of Petroleum and Minerals in Dhahran, Saudi Arabia, as an Assistant Professor. In 2002, he moved to the Department of Electrical Engineering, United Arab Emirates University, Al-Ain, UAE. In 2007, he joined the Electrical and Computer Engineering Program at Texas A&M University at Qatar, Doha, Qatar, where he is currently an Associate Professor. He published more than 100 refereed journal and conference papers and book chapters. He served as an Associate Editor and in technical committees of several international journals and conferences. His research interests include intelligent and adaptive control, control of time-delay systems, system biology, and system identification and estimation. Dr. Nounou is a senior member of IEEE.

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Mohamed Nounou (SM’08) received the B.S. degree (magna cum laude) from Texas A&M University, College Station, in 1995, and the M.S. and Ph.D. degrees from the Ohio State University, Columbus, in 1997 and 2000, respectively, all in Chemical Engineering. From 2000 to 2002, he was with PDF Solutions, a consulting company for the semiconductor industry, in San Jose, CA. In 2002, he joined the Department of Chemical and Petroleum Engineering at the United Arab Emirates University as an Assistant Professor. In 2006, he joined the Chemical Engineering Program at Texas A&M University at Qatar, Doha, Qatar, where he is currently an Associate Professor. He has published more than 80 refereed journal and conference papers and book chapters. He also served as an associate editor and in technical committees of several international journals and conferences. His research interests include process modeling and estimation, system biology, and intelligent control. He is a member of the American Institute of Chemical Engineers (AIChE) and a senior member of the IEEE.

Aniruddha Datta is a Professor and holder of the J. W. Runyon, Jr. ’35 Professorship II in the Department of Electrical and Computer Engineering at Texas A&M University. He received his Ph.D. degree in Electrical Engineering from the University of Southern California in 1991. His areas of interest include adaptive control, robust control, PID control and Genomic Signal Processing. He has authored or coauthored 5 books and over 100 journal and conference papers on these topics. He is a Fellow of IEEE.

Shankar P. Bhattacharyya (S’67-M’72-SM’86-F’89) was born in Yangon, Myanmar, on June 23, 1946. He received the B.Tech. degree from the Indian Institute of Technology (IIT), Bombay, in 1967 and the M.S. and Ph.D. degrees from Rice University, Houston, TX, in 1969 and 1971, respectively. He established the graduate program in automatic control at the Federal University, Rio de Janeiro, Brazil, from 1971 to 1980 and served as the Department Head of Electrical Engineering from 1978 to 1980. He has held an NRC-NASA Resident Research Associateship, a Senior Fulbright Lectureship, and the Boeing–Welliver Faculty Fellowship. At present, he is the Robert M. Kennedy Professor of Electrical Engineering at Texas A&M University, College Station. His contributions to control theory span 40 years and include the first solution of the linear servomechanism problem, the theory of robust and unknown input observers, a pole assignment algorithm based on Sylvester’s equation, the computation of the parametric stability margin, a generalization of Kharitonov’s theorem, the demonstration of the fragility of optimal and high order controllers, the synthesis of PID and fixed order controllers, and most recently an approach to model free, data driven controller synthesis. He has authored or coauthored five books and over 200 journal and conference papers. He is also a performing artist and has played concerts of North Indian Classical music on the Sarode, in several countries.